Abstract Algebra I: Master the Fundamentals of Algebraic Structures

Question 1

Which property is NOT necessarily true for a group?

Accepted Answer

Commutativity of the group operation

Answer

Closure under the group operation

Answer

Associativity of the group operation

Answer

Existence of an identity element

Question 2

Which property is NOT characteristic of a field?

Accepted Answer

Multiplicative inverse for every nonzero element

Answer

Commutative addition and multiplication

Answer

Associative addition and multiplication

Answer

Distributive property of multiplication over addition

Question 3

In abstract algebra, what property uniquely defines a field, distinguishing it from a ring?

Accepted Answer

Existence of multiplicative inverses for all non-zero elements

Answer

Existence of additive inverses for all elements

Answer

Commutativity of multiplication

Answer

Associativity of addition

Question 4

Which of the following accurately defines a group?

Accepted Answer

A set with an associative binary operation, an identity element, and inverses for all elements.

Answer

A set with an associative binary operation and an identity element, but not necessarily inverses.

Answer

A set with an associative binary operation and inverses for all elements, but not necessarily an identity element.

Answer

A set with an associative binary operation, but neither an identity element nor inverses.

Question 5

In group theory, which theorem states that the order of a subgroup divides the order of the group?

Accepted Answer

Lagrange's theorem

Answer

Cauchy's theorem

Answer

Sylow's theorem

Answer

Correspondence theorem

Question 6

Which property is NOT essential for a mathematical structure to be considered a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Answer

Inverse element

Question 7

Determine the type of group represented by (Z_6, +), where + denotes addition.

Accepted Answer

Cyclic group of order 6

Answer

Abelian group, not cyclic

Answer

Non-abelian group

Question 8

Which of the following is **not** a property of a group?

(a) Associativity
(b) Existence of an identity element
(c) Existence of inverses
(d) Closure under the group operation

Accepted Answer

(d) Closure under the group operation

Answer

(a) Associativity

Answer

(b) Existence of an identity element

Answer

(c) Existence of inverses

Question 9

What characteristic uniquely defines fields, setting them apart from rings?

Accepted Answer

Existence of additive and multiplicative inverses for all non-zero elements

Answer

Closure under addition and multiplication

Answer

Commutativity of addition and multiplication

Question 10

What is the defining characteristic that distinguishes a field from a ring?

Accepted Answer

A field has both additive and multiplicative inverses for every non-zero element.

Answer

A field has an additive identity element (0).

Answer

A field is closed under addition and multiplication.

Answer

A field contains a multiplicative identity element (1).

Question 11

In abstract algebra, which statement accurately describes the center of a group G?

Accepted Answer

It consistently exists as a normal subgroup.

Answer

It invariably forms a cyclic group.

Answer

It perpetually coincides with the commutator subgroup of G.

Question 12

Which property is **not** essential for a structure to be considered a group?

Accepted Answer

Commutativity of the operation

Answer

Associativity of the operation

Answer

Existence of an identity element

Answer

Existence of inverses for all elements

Question 13

Which property is NOT characteristic of a field?

Accepted Answer

The additive identity is unique.

Answer

The set of non-zero elements forms a multiplicative group.

Answer

Every non-zero element has a multiplicative inverse.

Question 14

Which defining property must a group possess?

Accepted Answer

Existence of an identity element

Answer

Commutativity of the group operation

Answer

Finiteness of the group

Answer

Abelian property of the group operation

Question 15

Consider a vector space V with dimension n and a subspace W of V with dimension m. Determine the dimension of the quotient space V/W.

Accepted Answer

n - m

Answer

n

Answer

m

Question 16

In a ring with unity and at least three distinct elements, why must there be a nilpotent element?

Accepted Answer

A nilpotent element is an element that equals its own power for some positive integer.

Answer

A nilpotent element is an element that squares to itself.

Answer

A nilpotent element is an element that cubes to itself.

Answer

A nilpotent element is an element that equals zero.

Question 17

Identify the option that is NOT a defining property of groups.

Accepted Answer

Commutativity of the group operation

Answer

Closure under the group operation

Answer

Associativity of the group operation

Answer

Existence of an identity element

Question 18

Which property ensures that, for any elements a, b, and c in a group, the operation (a * b) * c is equal to a * (b * c)?

Accepted Answer

Associativity

Answer

Closure

Answer

Identity element

Answer

Inverse element

Question 19

Consider the set G = {1, -1, i, -i} under the operation of multiplication. What is the order of this group?

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 20

Which of the following algebraic structures is a ring but not a field?

Accepted Answer

Integers

Answer

Rational numbers

Answer

Real numbers

Answer

Complex numbers

Question 21

Which of the following sets is a subspace of the vector space ℝ³?

Accepted Answer

{x ∈ ℝ³ | x₁ + x₂ + x₃ = 0}

Answer

{x ∈ ℝ³ | x₁, x₂, x₃ ≥ 0}

Answer

{x ∈ ℝ³ | x₁² + x₂² + x₃² = 1}

Answer

{x ∈ ℝ³ | x₁ is an odd integer}

Question 22

Which of the following is NOT a property of a group?

Accepted Answer

Commutativity of the operation

Answer

Closure under the operation

Answer

Associativity of the operation

Answer

Existence of an inverse element for each element of the group

Question 23

Let R be a commutative ring with unity. Which of the following is a field?

Accepted Answer

{a ∈ R : a is invertible} ⊂ R

Answer

{0, 1} ⊂ R

Answer

{a ∈ R : a² = a} ⊂ R

Answer

R

Question 24

Which of the following is an example of a non-abelian group?

Accepted Answer

The set of all permutations of a set

Answer

The set of all integers under addition

Answer

The set of all complex numbers under multiplication

Answer

The set of all real numbers under addition

Question 25

Within a group 'G', the centralizer of an element 'a' comprises which set of elements?

Accepted Answer

Elements of 'G' that commute with 'a', forming a subgroup under 'G's operation.

Answer

Normal subgroup of 'G'.

Answer

Cyclic subgroup generated by 'a'.

Answer

Abelian elements of 'G'.

Question 26

Which property is NOT characteristic of a group?

Accepted Answer

Distributive property

Answer

Closure under the operation

Answer

Associative property

Answer

Identity element

Question 27

Which of the following exemplifies a finite ring?

Accepted Answer

The set {0, 1, 2, ..., 10} under addition and multiplication modulo 11

Answer

The set of integers, denoted as ℤ

Answer

The set of complex numbers, denoted as ℂ

Answer

The set of all polynomials with real coefficients

Question 28

For a ring R with unity, which condition is essential for R to be classified as a field?

Accepted Answer

Every non-zero element of R has a multiplicative inverse

Answer

The ring operation is commutative

Answer

The ring contains a finite number of elements

Answer

The zero element of R is the sole element lacking an inverse

Question 29

The order of an element 'a' in a group is defined as:

Accepted Answer

The smallest positive integer 'n' such that 'a^n = e', where 'e' is the identity element

Answer

The number of elements in the group

Answer

The number of subgroups of the group

Answer

The number of generators of the group

Question 30

The center of a group is defined as:

Accepted Answer

The set of all elements that commute with every other element

Answer

The set of all elements that are self-inverse

Answer

The set of all subgroups of the group

Answer

The set of all generators of the group

Question 31

The polynomial ring over a field possesses which property?

Accepted Answer

It is a commutative ring

Answer

It is a non-commutative ring

Answer

It is a field

Answer

It is a division ring

Question 32

Which theorem in group theory establishes that every finite group can be represented within a permutation group?

Accepted Answer

Every finite group is isomorphic to a subgroup of a permutation group

Answer

Every group has a normal subgroup

Answer

Every Abelian group is isomorphic to a direct product of cyclic groups

Answer

Every ring is isomorphic to a field

Question 33

In a ring, which operation might not be associative?

Accepted Answer

Division

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 34

What mathematical concept preserves a group's operation?

Accepted Answer

A function that preserves the group operation

Answer

A function that preserves the ring operations

Answer

A function that preserves the field operations

Answer

A function that preserves the order of elements

Question 35

Which of the following is an example of a prime field?

Accepted Answer

The field of integers modulo 5

Answer

The field of rational numbers

Answer

The field of complex numbers

Answer

The field of all algebraic numbers

Question 36

In a finite-dimensional vector space, what is the relationship between the determinant of a linear transformation and its eigenvalues?

Accepted Answer

The determinant is the product of the eigenvalues.

Answer

The determinant is the sum of the eigenvalues.

Answer

The determinant is the absolute value of the largest eigenvalue.

Question 37

Which property guarantees that the outcome of an operation on three elements in a set remains unchanged regardless of the grouping of the elements?

Accepted Answer

Associativity

Answer

Commutativity

Answer

Distributivity

Question 38

In field theory, the characteristic of a field is defined as the smallest positive integer 'n' such that 'nx = 0' for all elements 'x' within the field. What is the term used to describe this concept?

Accepted Answer

Characteristic

Answer

Rank

Answer

Degree

Answer

Exponent

Question 39

Consider the set G = {1, -1, i, -i} under the operation of multiplication. Which of the following statements is true about G?

Accepted Answer

G satisfies the associative property for the given operation.

Accepted Answer

Every element in G has an inverse under the given operation.

Answer

G is not closed under the given operation.

Answer

G does not have an identity element.

Question 40

In a group, the order of an element 'a' is defined as:

Accepted Answer

The smallest positive integer 'n' such that a^n = e, where 'e' is the identity element

Answer

The largest positive integer 'n' such that a^n = e, where 'e' is the identity element

Answer

The number of distinct elements in the group generated by 'a'

Question 41

Which property characterizes free groups?

Accepted Answer

They have no non-trivial relations among their generators.

Answer

They are always finitely generated.

Answer

The free group on two generators is isomorphic to the alternating group on three elements.

Question 42

Which of the following is NOT a defining property of a group?

Accepted Answer

Commutativity

Answer

Existence of an identity element

Answer

Associativity of the operation

Answer

Existence of inverse elements for all elements

Question 43

Which of the following is an example of a field?

Accepted Answer

The set of rational numbers Q

Answer

The set of matrices of size 2 x 2 with real entries and determinant 1

Answer

The set of real numbers without 0

Answer

The set of integers Z

Question 44

Which of the following is a homomorphism from the group (Z, +) to the group (Z_6, +)?

Accepted Answer

f(x) = x mod 6

Answer

f(x) = x^2 mod 6

Answer

f(x) = 2x mod 6

Question 45

Which of the following is equivalent to the statement 'a divides b in the ring Z' ?

Accepted Answer

There exists an element c in Z such that b = ac

Answer

a and b are relatively prime

Answer

a is a factor of b

Question 46

Consider the field extension F/K. Which of the following statements is TRUE if F is a finite field?

Accepted Answer

[F:K] is finite.

Answer

[F:K] can be either finite or infinite.

Answer

[F:K] is infinite.

Question 47

Let R be a ring with unity. Which of the following is a sufficient condition for R to be a field?

Accepted Answer

Every nonzero element of R has a multiplicative inverse

Answer

R is commutative and associative

Answer

R has no zero divisors

Question 48

Which of the following is a cyclic group?

Accepted Answer

<Z10, +>

Answer

<S10, *>

Answer

<A5, *>

Answer

<Z10, *>

Question 49

Which of the following is a free group?

Accepted Answer

The group consisting of all finite strings of symbols chosen from an alphabet

Answer

The additive group of integers

Answer

The group of permutations of a finite set

Question 50

Which of the following is a normal subgroup of the symmetric group S3?

Accepted Answer

The alternating group A3

Answer

The cyclic group of order 2

Answer

The cyclic group of order 3

Question 51

Which of the following is NOT a property of a group?

Accepted Answer

The operation is commutative.

Answer

For each element, there exists an inverse element.

Answer

The operation is associative.

Answer

There exists an identity element.

Question 52

Which of the following sets forms a group under matrix multiplication?

Accepted Answer

{[1 0], [0 1], [-1 0], [0 -1]}

Answer

{[1 0], [0 1], [1 1]}

Answer

{[1 0], [0 1], [0 0]}

Answer

{[1 0], [0 1], [1 -1]}

Question 53

Which of the following is an example of a field?

Accepted Answer

The set of rational numbers with usual addition and multiplication.

Answer

The set of even integers with usual addition and multiplication.

Answer

The set of integers with usual addition and multiplication.

Question 54

Let R be a ring and I be an ideal of R. Which of the following is a necessary condition for I to be a maximal ideal?

Accepted Answer

R/I is a field.

Answer

I is a prime ideal.

Answer

I is a principal ideal.

Answer

R/I is a ring with unity.

Question 55

Let f: G → H be a homomorphism between groups G and H. Which of the following is true?

Accepted Answer

f(e_G) = e_H, where e_G and e_H are the identity elements of G and H respectively.

Answer

f is always a bijection.

Answer

f(a * b) = f(a) + f(b) for all a, b in G.

Question 56

Consider the polynomial ring Z[x]. What is the degree of the polynomial 3x³ + 2x - 1?

Accepted Answer

3

Answer

1

Answer

0

Answer

2

Question 57

What is the order of the cyclic group generated by the element (1 2 3) in the symmetric group S₃?

Accepted Answer

3

Answer

6

Answer

2

Answer

1

Question 58

Which of the following is NOT a property of a field?

Accepted Answer

Every non-zero element has infinitely many inverses.

Answer

Every non-zero element has an inverse.

Answer

It is a commutative ring with unity.

Question 59

Which of the following is NOT a property of a group?

Accepted Answer

Commutativity

Answer

Identity element

Answer

Associativity

Answer

Inverse element

Question 60

What is the order of the cyclic group generated by the element a with order 6?

Accepted Answer

6

Answer

24

Answer

12

Answer

3

Question 61

Which of the following is an example of a homomorphism between groups?

Accepted Answer

A function that preserves the group operation.

Answer

A function that is injective.

Answer

A function that maps elements of one group to another group.

Answer

A function that is one-to-one and onto.

Question 62

What is the kernel of the homomorphism f: Z -> Z_4 defined by f(n) = [n]_4?

Accepted Answer

2Z

Answer

3Z

Answer

Z

Question 63

Which of the following is a consequence of the Sylow Theorems?

Accepted Answer

If a group has a prime number of elements, it is cyclic.

Answer

Every group of order n has a unique Sylow p-subgroup for every prime divisor p of n.

Answer

If a group has a composite order, it has a normal subgroup of prime order.

Question 64

Let F be a field and V a vector space over F. Which of the following is a subspace of V?

Accepted Answer

The set of all vectors that are multiples of a fixed vector.

Answer

The set of all vectors that are perpendicular to a fixed vector.

Answer

The set of all vectors that have the same length.